Prove that subgroups and quotient groups of nilpotent groups are nilpotent. (Your proof should work for infinite groups.) Give an explicit example of a group which possesses a normal subgroup such that and are nilpotent but is not nilpotent.

We showed that subgroups and quotients of nilpotent groups are nilpotent in the lemmas to this previous exercise.

Consider now . is a normal subgroup of order 3, and thus is abelian, hence nilpotent. Moreover, is abelian, hence nilpotent. However, has distinct Sylow 2-subgroups and (for example). By Theorem 3, is not nilpotent.